RF spectrum analysis of optical signals is typically performed using electronic means, such as a fast photodetector and a super-heterodyne RF spectrum analyzer. This approach, however, is limited in terms of bandwidth (e.g. due to the bandwidth limitations of the photodetector).
It has been determined that improved RF spectrum analysis of optical signals can be obtained using nonlinear optics. Using this approach, a source under test, with electric field E(t), is coupled with a monochromatic source at an optical frequency ω0, and sent into a nonlinear medium with third order nonlinear response, including cross phase modulation (XPM), cross gain modulation (XGM) and two-photon absorption (TPA).
The output electric field around the frequency of the monochromatic source is related to the input by:E′(t)=√{square root over (I0)}exp(−iω0t)exp[αI(t)];
where α describes the effect of the nonlinear interaction of the source under test on the monochromatic source.
If the modulation due to the nonlinear interaction is small compared to one, one has exp [αI(t)]=1+αI(t), which yields:|∫E′(t)exp[iαx]dt|2=I0δ(ω−ω0)+I0|α2||∫I(t)exp[i(ω−ω0)t]dt|2.
It therefore appears that the output optical spectrum, measured for example using a grating-based optical spectrum analyzer (OSA) or a scanning Fabry-Perot etalon followed by a photodetector, is representative, up to a Dirac function at the optical frequency ω0, of the RF spectrum of the source under test centered at ω0. However, most nonlinear interactions are polarization-dependent. The previous equations assume that the source under test and the monochromatic laser have the same state of polarization in the nonlinear medium. This, however, can be difficult to achieve for several reasons. First, the source under test has, in most cases, an unknown polarization state. Secondly, the polarization state of the source under test can be time-dependent, for example, when pulses are polarization multiplexed so that adjacent temporal bits do not interfere. Thirdly, the polarization state of the two sources can be modified during propagation in the nonlinear element because of, for example, polarization mode dispersion in a non-polarization maintaining fiber. Finally, and perhaps more importantly, the definition of the RF spectrum is polarization-independent, i.e. for a source under test with electric field E(t)=Ex(t){circumflex over (x)}+Ey(t)ŷ, the RF spectrum is:S(Ω)=|∫(Ix(t)+Iy(t))·exp(iΩt)dt|2 where Ix(t)=|Ex(t)|2 and Iy(t)=|Ey(t)|2. If XPM in a fiber is used, the nonlinear coefficients describing the modulation of the monochromatic laser by the source under test depend upon the relative state of polarization of the two sources. Therefore, no correspondence between the RF spectrum and the optical spectrum after modulation is obtained in the general case for such implementation since the measured optical spectrum does not depend identically upon the polarized components of the source under test along {circumflex over (x)} and ŷ.
There is therefore a need for method and apparatus to improve on this technique and obtain polarization-independent determination of the RF spectrum, while retaining the capability to provide ultrahigh bandwidth RF spectrum analysis.